 Finite-field Bogolyubov lemma

If $A\subseteq {\mathbb{F}}_p^n$ has density $\alpha$, then $2A-2A$ contains a vector subspace of codimension at most $2{\alpha }^{-2}$. It is the annihilator of the Fourier spectrum on which $|\widehat{1_A}|$ is at least ${\alpha }^{3/2}/\sqrt{2}$.